The existence claim for the singular value decomposition (SVD) is quite strong: “Every matrix is diagonal, provided one uses the proper bases for the domain and range spaces” (Trefethen & Bau III, 1997). MIT professor Gilbert Strang has a wonderful lecture on the SVD, and he includes an existence proof for the SVD. The goal of this post is to force myself to walk through his proof carefully. For legibility, I break the proof into two sections: an overview and details. I hope this allows the read...